SEDIMENT CONCENTRATIONS AND TRANSPORT DUE TO NEAR-BED JET FLOW: STJET-MODEL

Transcription

1 SEDIMENT CONCENTRATIONS AND TRANSPORT DUE TO NEAR-BED JET FLOW: STJET-MODEL 1. Introduction This note describes the STjet-model, which computes the sediment concentrations and transport generated by a jet-type flow near the bed consisting of mud, silt and sand. A background tide-riven and/or winddriven flow may be present or not. 2. Hydrodynamics The vertical distribution of the velocity is represented, as: u z = u max,b (z/δ) 1/7 + [u mean/(-1+ln(h/z o))] ln(z/z o) (2.1) with: u max,b= peak velocity of near-bed jet, z= level above bed (m), δ= layer thickness of jet, h= water depth (m), u mean= depth-averaged tide-driven and/or wind-driven velocity, z o= 0.033k s,c = zero-velocity level, k s,c = current-related bed roughness. Figure 2.1 shows an example of near-bed jet-type flow near the bed. Height above bed (m) Velocity profile Horizontal near-bed velocity (m/s) Figure 2.1 Flow velocity profile (u max = 1 m/s, δ = 0.5 m, u mean = 0.2 m/s, h = 150 m; k s,c = 0.01 m) The bed-shear stress is given by: τ b = ρ w (u *) 2 = ρ [κ u z=0.5δ/ln(15δ/k s,c)] 2 (2.2) with: τ b = bed-shear stress, ρ w = fluid density, u *= bed-shear velocity, κ=0.4, The fluid mixing is given by: ε z= [[ε max,b {1-(1-2z/(2δ)) 2 ] 2 + [ε max,h {1-(1-2z/(h)} 2 ] 2 ] 0.5 for z < 2δ (2.3) ε z= ε max,h [1-(1-2z/(h)) 2 ] for 2δ < z < 0.5h (2.4) ε z= ε max,h = 0.1 u *m h for z > 0.5h (2.5) 1

2 with: ε= mixing coefficient, ε max,b = 0.05 u max δ = maximum mixing coefficient near the bed due the jet flow, u *m= bed-shear velocity due to the mean current only (= u mean g 0.5 /C), C= Chézy-coefficient= 5.75g 0.5 log(12h/k s,c). Based on linear wave theory, the near-bed velocity due to oscillatory flow (waves) is given by: u w= π H s/(t p sinh(kh) (2.6) with: H s= significnt wave height, T p= peak wave period, h= water depth, k=2π/l, L= wave length. The bed-shear stress due to oscillatory flow is given by: τ b,w = ρ w (u *,w) 2 = 0.25 ρ f w (u w) 2 (2.7) f w = exp(-6+5.2(a w/k s,w)) (2.8) with: u w= =peak orbital velocity (linear wave theory), A w= T p/(2π)u w =peak orbital excursion, T p= peak wave period, f w= wave-related friction coefficient, k s,w= wave-related roughness. The effective bed-shear stresses for sediment transport are represented as: / τ b,c = µ c τ b,c (2.9) τ b,w/ = µ w τ b,w (2.10) µ c= f c/ /f c= current-related efficiency factor, f c= 0.24/(log(12h/k s,c)) 2 = current-related friction coefficient, f c/ = 0.24/(log(12h/d 90)) 2 =grain-related friction coefficient, d 90= grain size of sediment mixture. µ w= 0.7/D *= wave-related efficiency factor (µ w,min=0.14, µ w,max=0.35). 3. Sediment concentrations and transport 3.1 Definitions In most tidal basins the sediment bed consists of a mixture of sand, silt and mud. The sand-silt-mud mixture generally behaves as a mixture with cohesive properties when the mud fraction is dominant (p mud>0.3) and as a non-cohesive mixture when the sand fraction is dominant (p sand>0.7). The distinction between non-cohesive mixtures and cohesive mixtures can be related to a critical mud content (p mud,cr). Most important is the value of the clay/lutum-fraction (sediments<16 µm) in the mixture. Cohesive properties become dominant when the clay-fraction is larger than about 0.05 to 0.1. Three sediment fractions are distinguished, as follows: mud; particles smaller than 16 µm with cohesive properties; silt; particles between 16 and 63 µm with non-cohesive properties; sand; particles larger than 63 µm with non-cohesive properties. The velocities and sand concentrations are computed as a function of height (z) above bed. The grid points over the depth (50 points) are distributed according to an exponential function, as follows: z=a[h/a] (k-1)/(n-1) (3.1) with: a= reference height above bed (input value), h= water depth between bed and water surface, k= index number of point k, N= total number of grid points (=50). 2

3 Near the bed the sediment concentrations are strongly decreasing in vertical direction resulting in stratification effects as the fluid density will decrease in vertical direction. Stratification effects will result in the damping of turbulence because turbulence energy is consumed in the mixing of heavier fluid from a lower level to a higher level against the action of gravity. The usual method to account for the stratification effect on the velocity and concentration profiles is the reduction of the fluid mixing coefficient by introducing a damping factor related to the Richardson-number (Ri), as follows: ε= φ ε o with ε o=fluid mixing coefficient in water without sediment, φ= F(Ri) = damping factor (< 1), Ri = local Richardson number. 3.2 Sediment concentration equations The vertical distribution of the sediment concentrations in uniform flow can be described by: c w s + ε dc/dz = 0 (3.2) dc/dz= -(w s/ε s) c (3.3) c z = c z- z + c = c z- z - (w s/ε s) c z = c z- z [1 -(w s/ε s) z] (3.4) with: w s= settling velocity of sediment, ε s= φ ε= sediment mixing coeficient, φ= turbulence damping coefficient. The Richardson number (Ri), which expresses the near-bed stratification effects close to the sediment bed, is defined as follows: Ri= [-(g/ρ)][dρ/dz]/[(du/dz) 2 ]= [-(ρ s-ρ w)g/((ρ w+(ρ s-ρ w)c)][dc/dz]/[(du/dz) 2 ] (3.5) with: ρ= fluid-sediment mixture density=ρ sc+(1-c)ρ w, c= volume sediment concentration, ρ w= fluid density, ρ s= sediment density. The concentration and velocity gradients are determined by using the values at heights z k-1 and z k-2. The damping function is expressed as: φ= (1+α d Ri 0.5 ) -1 (3.6) with: α d= input value (in range of 5 to 20), Ri=Richardson number. The reference concentration c a of the sediment fractions is described by: c a=0.015 α ca (p i) f silt,i (d m,i/a) (T i) 1.5 (D *,i) -0.3 (3.7) with: α ca= scaling coefficient (default=1), p i= sediment fraction ( p i=1; three fractions), d m,i= mean particle size of sediment fraction i, D *,i= d m,i[(s-1)g/ν 2 ] = dimensionless particle parameter of sediment fraction i, f silt,i=d sand/d m,i (with f silt=1 for d sand/d m,i<1), d sand= m (smallest sand size), T i=(τ b,cw/ -χτ b,cr,i)/χτ b,cr,i= dimensionless bed-shear stress parameter, τ b,cr,i= critical bed-shear stress of sediment fraction i, τ b,cw/ = µ cτ b,c+µ wτ b,w= effective bed-shear stress due to current and waves, s=ρ s/ρ= relative density, χ= factor representing the effects of mud on the critical shear stress of silt and sand fraction i, ν= kinematic viscosity coefficient, ρ s= sediment density, ρ= fluid density. 3

4 3.3 Settling velocity Mud The concentration-dependent settling velocity of the mud fraction is represented as: w mud = exp[α 1 ln(c) + α 2 - α 3]; for flocculation range c (3.8) α 1= ln(w mud,max/w mud,min) α 2= 2.09 ln(w mud,max) α 3=1.09 ln(w mud,min) w mud= w mud,max(1-c) 4 for hindered settling range c > (3.9) with: w mud,max= maximum settling velocity at c= (w mud,max = to m/s or 0.5 to 3 mm/s; input value), w mud,min= minimum settling velocity at c= (w mud,min= to m/s or 0.05 to 0.1 mm/s; input value). The settling velocity at height z k is determined by using the concentration values at height z k Settling velocity (m/s) concentration-dependent settling velocity Volume concentration (-) Figure 3.1 Settling velocity as function of volume concentration; flocculation range for c< and hindered settling range for c>0.0025; w mud,max=0.001 m/s, w mud,min= m/s Some characteristic values of the settling velocity w mud are: c= (=6.6 kg/m 3 ) w mud=0.001 m/s = 1 mm/s c= (=0.66 kg/m 3 ) w mud= m/s = 0.38 mm/s c= (=0.066 kg/m 3 ) w mud= m/s = mm/s c=0.025 (=66 kg/m 3 ) w mud= m/s = 0.9 mm/s c=0.1(=265 kg/m 3 ) w mud= m/s = 0.66 mm/s Equation (3.8) using w mud,max=0.001 m/s and w mud,min= m/s, is shown in Figure 3.1. Silt and sand The settling velocity of the silt and sand fractions are assumed to be constant (not dependent on concentration). 4

5 3.4 Suspended load and bed load transport The suspended sand transport q s can be computed by integration of the product of velocity and concentration over the water depth: q s= a h (u c) dz (3.10) The bed load transport (in kg/m/s) can be computed as: q b= c a u a a (3.11) with: c a= near-bed concentration (see Equation 3.7), u a= flow velocity a level z=a. The total load transport is computed as: q tot=q s+q b (3.12) 5

6 4 Example computation Table 4.1 shows an example of the model input parameters. INPUT VALUES Water depth h 150 (m) Depth-averaged velocity Umean 0.2 (m/s) Peak water velocity near-bed jet flow Umax,near bed 1 (m/s) Layer thickness of near bed jet flow Delta 0.5 (m) Wave height Hs 0 (m) Wave period Tp 15 (s) Salinity-induced density gradient (max. value 0.001) dρ/dx 0 (kg/m3/m) Coeff. salinity-induced flow (0.5 to 2; larger yields smaller effect) Gamma 1 (-) Percentage of MUD<16 um pmud 0.2 (-) Percentage of SILT (16 to 63 um) psilt 0.3 Percentage of SAND >63 um psand 0.5 (-) Mean size of MUD fraction <16 um dm,mud (m) Mean size of SILT fraction 16 to 63 um dm,silt (m) Mean size of SAND fraction >63 um dm,sand (m) Selection of MUD settling velocity; 1=constant=ws,max, 2=non-constant 2 Maximum MUD settling velocity at conc= ws,max,mud (m/s) Minimum MUD settling velocity at c= ws,min,mud (m/s) Settling velocity of SILT fraction ws,silt (m/s) Settling velocity of SAND fraction ws,sand (m/s) Critical bed-shear stress of MUD fraction tb,cr,mud 0.5 (N/m2) Critical bed-shear stress of SILT fraction tb,cr,silt 0.3 (N/m2) Critical bed-shear stress of SAND fraction tb,cr,sand 0.3 (N/m2) Sediment density rhos 2650 (kg/m3) Fluid density seawater rhow 1025 (kg/m3) Fluid viscosity nu (m2/s) Bed roughness for shear stress/mixing-currents ks,c, bed surface 0.01 (m) Bed roughness for shear stress/mixing-waves ks,w,bed surface 0.01 (m) Effective roughness for velocity profile ks.c, velocity profile 0.01 (m) Reference level a 0.01 (m) Scaling Coefficient reference concentration MUD 1 (-) Scaling Coefficient reference concentration SILT 1 (-) Scaling Coefficient reference concentration SAND 1 (-) Coefficient turbulence damping 5 (-) Table 4.1 Input data STjet-model Figure 4.1 shows the results of an example computation based on the data of Table 4.1. The bed consists of sand (50%), silt (30% and mud (20%). A jet-type flow near the bed is present with maximum velocity u max= 1m/s over a tickness of δ= 0.5 m. The background flow in water depth of h= 150 m has a mean velocity of u mean=0.2 m/s. The near-bed sediment concentrations are in the range of 1 to 10 kg/m 3. The sand fraction is suspended up to a level of 0.25 m above the bed; so mainly bed laod transport. The silt fraction is suspended up to a level of 1 m above the bed. The mud fraction with concentrations larger than 10 mg/l (0.01 kg/m 3 ) is suspended up to a level of 20 m above the bed. The mixing effect of the weak background flow is too small to generate significant mud concentrations above 20 m. 6

7 Mud concentration (<16 um) Silt concentration (16 to 63 um) Sand concentration (> 63 um) Velocity profile Height above bed (m) Mud transport Silt transport Sand transport Cconcentration (kg/m3), Velocity (m/s), Transport (kg/m2/s) Figure 4.1 Sediment concentrations and transport of example computation 7